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30Spline

# 30Spline - Nonparametric regression using smoothing splines...

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Nonparametric regression using smoothing splines Smoothing is fitting a smooth curve to data in a scatterplot Will focus on two variables: Y and one X Our model: y i = f ( x i ) + ε i , where ε 1 , ε 1 , . . . ε n are independent with mean 0 f is some unknown smooth function Up to now f has a specified form with unknown parameters f could be linear or nonlinear in the parameters, functional form always specified If f not determined by the subject matter, we may prefer to let the data suggest a function form c 2011 Dept. Statistics (Iowa State University) Stat 511 section 30 1 / 50 Why estimate f ? can see features of the relationship between X and Y that are obscured by error variation summarizes the relationship between X and Y provide a diagnostic for a presumed parametric form Example: Diabetes data set in Hastie and Tibshirani’s book Generalized Additive Models Examine relationship between age of diagnosis of diabetes and log of the serum C-peptide concentration Here’s what happens if we fit increasing orders of polynomial, then fit an estimated f c 2011 Dept. Statistics (Iowa State University) Stat 511 section 30 2 / 50 5 10 15 3.0 4.0 5.0 6.0 Age at Diagnosis log C-peptide concentration c 2011 Dept. Statistics (Iowa State University) Stat 511 section 30 3 / 50 5 10 15 3.0 4.0 5.0 6.0 Age at Diagnosis log C-peptide concentration linear fit c 2011 Dept. Statistics (Iowa State University) Stat 511 section 30 4 / 50

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30Spline - Nonparametric regression using smoothing splines...

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